The Ura characters never appear in the anime but they're in some of the Corner Shop games and manga.
Mini-series
TV de Hakken! Tamagotch - A series of shorts based on the vintage Tamagotchis from the late 90s. The characters don't talk, just make little sounds.
Let's Go Tamagotchi - A simple YouTube mini-series about Mametchi, Memetchi, Kuchipatchi, and their classmates.
Full TV anime
Tamagotchi! - Composed of over 140 episodes, this is about the daily life of Tamagotchi characters and as the episodes go on, more characters and plot elements are introduced. Around the middle of the series or so, Lovelitchi and Melodytchi go around collecting Tama Hearts, which form from a bond between two characters.
Tamagotchi! Yume Kira Dream - Featuring the characters from the Tamagotchi P's, Yumemitchi and Kiraritchi are given a magical bag that can transform them and give them special talents. However, they must keep it a secret or else their dream will never come true.
Tamagotchi! Miracle Friends - Miraitchi and Clulutchi, twin sisters from the future, get sent to present-day and must collect all 8 Dreambakutchis to help them return to their own timeline. However, a mysterious Tamagotchi called X-Kamen/Smarttchi wants to obtain the Dreambakutchis too.
Go-Go Tamagotchi - A series where Tamagotchi Town and Dream Town combine, following the escapades of all the characters from both towns interacting with eachother. The last Tamagotchi anime as of now.